sdata <- read.table("Sdata.csv", sep=" ", header=TRUE)
pairs(sdata)
plot(Y ~ X13, data=sdata)
lm1 <- lm(Y ~ X13 + I(X13^2), data=sdata)
b <- coef(lm1)
curve(b[1] + b[2]*x + b[3]*x^2, add=TRUE)
summary(lm1)
##
## Call:
## lm(formula = Y ~ X13 + I(X13^2), data = sdata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -36.040 -15.731 -3.089 6.602 69.116
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 33.79442 4.76248 7.096 2.11e-08 ***
## X13 -1.28604 0.55486 -2.318 0.0261 *
## I(X13^2) 17.05763 0.03903 437.021 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 22.94 on 37 degrees of freedom
## Multiple R-squared: 0.9999, Adjusted R-squared: 0.9999
## F-statistic: 1.727e+05 on 2 and 37 DF, p-value: < 2.2e-16
pairs(cbind(R=lm1$res, sdata))
lm2 <- lm(Y ~ X13 + I(X13^2) + X2, data=sdata)
par(mfrow=c(1,2))
plot(lm2, which=1:2)
pairs(cbind(R=lm2$res, sdata), col=as.factor(sdata$X10))
lm3 <- lm(Y ~ X13 + I(X13^2) + X2 + I(X10==2) + I(X10==3) + I(X10==4), data=sdata)
summary(lm3)
##
## Call:
## lm(formula = Y ~ X13 + I(X13^2) + X2 + I(X10 == 2) + I(X10 ==
## 3) + I(X10 == 4), data = sdata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.324 -2.436 -0.216 2.007 17.467
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.060392 2.455816 -2.468 0.01895 *
## X13 -0.299121 0.132282 -2.261 0.03046 *
## I(X13^2) 17.011411 0.009588 1774.169 < 2e-16 ***
## X2 5.673183 0.227535 24.933 < 2e-16 ***
## I(X10 == 2)TRUE 9.334707 2.713169 3.441 0.00159 **
## I(X10 == 3)TRUE -5.802909 2.489933 -2.331 0.02604 *
## I(X10 == 4)TRUE -0.030425 2.515469 -0.012 0.99042
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.205 on 33 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.118e+06 on 6 and 33 DF, p-value: < 2.2e-16
par(mfrow=c(1,2))
plot(lm3, which=1:2)
lm4 <- lm(Y ~ X13 + I(X13^2) + X2 + I(X10==2) + I(X10==3), data=sdata)
summary(lm4)
##
## Call:
## lm(formula = Y ~ X13 + I(X13^2) + X2 + I(X10 == 2) + I(X10 ==
## 3), data = sdata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.3237 -2.4370 -0.2274 2.0080 17.4634
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.076582 2.028434 -2.996 0.005082 **
## X13 -0.299190 0.130201 -2.298 0.027845 *
## I(X13^2) 17.011443 0.009064 1876.803 < 2e-16 ***
## X2 5.672467 0.216446 26.207 < 2e-16 ***
## I(X10 == 2)TRUE 9.353792 2.174441 4.302 0.000135 ***
## I(X10 == 3)TRUE -5.784409 1.935662 -2.988 0.005179 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.128 on 34 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.382e+06 on 5 and 34 DF, p-value: < 2.2e-16
par(mfrow=c(1,2))
plot(lm4, which=1:2)
lm5 <- lm(Y ~ X13 + I(X13^2) + X2 + I(X10==2) + I(X10==3) + I(X10==2):X2, data=sdata)
summary(lm5)
##
## Call:
## lm(formula = Y ~ X13 + I(X13^2) + X2 + I(X10 == 2) + I(X10 ==
## 3) + I(X10 == 2):X2, data = sdata)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1183 -1.7709 -0.1383 1.6594 18.8668
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -5.383238 2.038060 -2.641 0.0125 *
## X13 -0.234769 0.134240 -1.749 0.0896 .
## I(X13^2) 17.007982 0.009162 1856.452 <2e-16 ***
## X2 5.581265 0.220190 25.348 <2e-16 ***
## I(X10 == 2)TRUE 2.498485 4.918242 0.508 0.6148
## I(X10 == 3)TRUE -5.892473 1.898500 -3.104 0.0039 **
## X2:I(X10 == 2)TRUE 1.030564 0.666338 1.547 0.1315
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.026 on 33 degrees of freedom
## Multiple R-squared: 1, Adjusted R-squared: 1
## F-statistic: 1.199e+06 on 6 and 33 DF, p-value: < 2.2e-16
par(mfrow=c(1,2))
plot(lm5, which=1:2)
\[ Y_i = \beta_0 + \beta_1 X_{13} + \beta_2 X_{13}^2 + \beta_3 X_2 + \beta_4 X_{10==2} + \beta_5 X_{10==3} + \epsilon_i \] Our estimates for the \(\beta\)’s in the above model are as follows:
confint(lm4, level=.99)
## 0.5 % 99.5 %
## (Intercept) -11.6109499 -0.54221480
## X13 -0.6544286 0.05604804
## I(X13^2) 16.9867131 17.03617369
## X2 5.0819184 6.26301577
## I(X10 == 2)TRUE 3.4210579 15.28652562
## I(X10 == 3)TRUE -11.0656571 -0.50316057
| Term | Estimate | Lower Bound | Upper Bound |
|---|---|---|---|
| \(\beta_0\) | -6.076582 | ||
| \(\beta_1\) | -0.299190 |
pairs(cbind(R=lm4$res, sdata), col=as.factor(sdata$X11))
pairs(cbind(R=lm4$res, sdata), col=as.factor(sdata$X12))
pairs(cbind(R=lm4$res, sdata), col=as.factor(sdata$X10))